The Lone Star State senator's amendment flies in the face of the bipartisan immigration reform measure that would allow possible citizenship.

He's also called for prohibiting federal, state, or local entitlement benefits for those in the country illegally.

Flores said while he has not read all of the specifics of Cruz's plan, "the underlying issue is not whether or not the folks are here to become citizens.

"They're really just here for a job for the most part. So if they square themselves with the law, if they reconcile themselves with the laws that they've broken and if they pay their back taxes and they pay their fines and so forth, really, all they want is the opportunity to have a job.''

He said if that can be done in such a way where "we're not displacing indigenous Americans, then we're probably okay in doing this.

"Most of them didn't come here for citizenship. So why should we talking about a path to citizenship?''

On Thursday, the Judiciary Committee rejected a Republican attempt to significantly delay the legalization process for 11 million undocumented immigrants.

That process is one of the main focuses of the non-partisan bill by the so-called "Gang of 8'' senators.